Importance of Tollmien's Counter Example

Efforts to construct a general theoretical basis containing the essential features of Tollmien's counter example to the sufficiency of Rayleigh's theorem on point of inflexion have resulted in the determination of a pair of upper bounds of the rate of growth of arbitrary unstable disturbances; whereas, the necessary condition of the existence of these upper bounds have provided access to a sufficient condition of stability in its simplest form in the equilibrium of homogeneous incompressible inviscid parallel shear flows that are not known as yet and go beyond the works of Rayleigh 1], Tollmien 2], Friedrichs 3], Fjortoft 4], Hoiland 5], Howard 6, 7], Hickernell 8], and Banerjee et al. 9]. An alternative proof of the result that a wide class of such flows could be made stable by bringing the boundaries sufficiently close, although the flow has a point of inflexion inside the domain of flow with the Fjortoft's criterion satisfied, which is derived by Drazin and Howard 10] from variational formulation of the problem follows as an outcome of the expressions of these upper bounds. The counter example has played the role of a forerunner for much of the development that followed in its wake after 1935, and the present succession of papers is especially undertaken to investigate the trail left behind by the counter example and, it is hoped, to arrive at a necessary and sufficient condition of stability in its simplest form, which is still missing in the literature on the subject.